The less restricted the air intake system and exhaust system the quicker the turbos will spool. So free up your factory air intake with one of the multiple options out there. For the exhaust getting rid of the cats is the first thing I'd do. The single biggest item to perceptably reducing time to spool will be the catless downpipes though.

Pushing the M button certainly makes things more lively, but to me the biggest effect of the M button feels like it changes the gain of the throttle pedal vs throttle body position. Regardless, throttle response lag just after tip-in is still present.

I think the issue is spool time. With a DCI this becomes really apparent...push the throttle, hear the spool, then get pushed back.

Maybe it's simply a turbo thing, and I'm used to very responsive normally aspirated motors. My E30 runs on Megasquirt, my 540i/6 has a lot of displacement. Both cars blow the 1M out of the water when it comes to what happens 100-200 ms after you hit the gas.

Pushing the M button certainly makes things more lively, but to me the biggest effect of the M button feels like it changes the gain of the throttle pedal vs throttle body position. Regardless, throttle response lag just after tip-in is still present.

I think the issue is spool time. With a DCI this becomes really apparent...push the throttle, hear the spool, then get pushed back.

I suspect that is a placebo effect.

The stock intake effectively "muffles" the sound of the intake tract and turbos, so you're hearing them much more with the DCI's.

Since DCIs offer gains of only 4-6 rwhp, I don't believe they're causing the turbos to spin-up more quickly and suspect that you're really just hearing them more.

Shirley you can't be serious. Are you talking track only or the street as well.

Track only but even then I don't think it's much benefit on the 1M. I was mostly kidding really. It's an old technique 911 turbo guys used to keep the turbo spooled back when big lag was an issue. They would left foot brake and use the right foot to work the throttle so corner exit wasn't a big waiting game.
When I tracked my 1M I didn't think there was any delay in boost worth mentioning. When power oversteer at corner exit is a common characteristic of the 1M I don't see where said delay is. Like others have said, the M button helps and it's just a push away. I preferred to leave it off because I like 100% throttle to be 100% throttle. That's me and I'm old school.